
The superradiant instability happens only at \(M\omega _R^{nlm}< m/2\)  (see Fig. REF ), in which region \(a^{nlm}_{*C}\)  is an increasing function of \(\omega _R^{nlm}\) , and consequently an increasing function of \(n\) . Therefore, fixing \(l\)  and \(m\) , the mode with a larger value of \(n\)  reaches the superradiance condition earlier. One can further conclude that the \(\lbrace n,l,m\rbrace \)  modes with \(n\ge 2\)  is less important since it has a smaller \(\omega _I\)  and a shorter growing time. A caveat is that one could always make the large \(n\)  modes dominant by setting their initial masses several orders of magnitude larger than the small \(n\)  modes. This initial condition may happen if a BH passes by a scalar-dense area. We do not consider this possibility in this work. A relevant discussion of the dependence on the initial condensate mass can be found in Ref. [1]}.
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